750 Mr. H. Bateman on some Problems 



We may conclude from these inequalities that 



lis I2? I3) • • • ■ 



form a decreasing set of quantities ; we have finally to find 

 whether 



Now it follows at once from the equations * 

 I n (V) = —1 cosh (y cos <b)d<b, 



Ij (»/)=• — I sinh(vcos<£) cos (f>d(j), 

 77 \ 

 that 



I 1 «<IoW; 



for clearly 



sinh (v cos<£) cos cjx cosh (v cos (£) 



for all values of <f>. Hence r — is the most probable value 

 of r. 



4. It is also of some interest to find the value of v for which 

 the chance 



e-%(v) 



is a maximum when r is given. To do this we have to solve 

 the equation 



Putting X n = " we easily find from the recurrence 



formulae n ^ v ^ 



T ' ■ n T -T 

 T / n~l T _ T 



1 71-1 ~ J-71-1 - L ? Z5 



that 



(x. + 3(^-==i).i. 



* Cf. Whittaker's l Analysis,' p. 307. 



